Recent deep image compression methods have achieved prominent progress by using nonlinear modeling and powerful representation capabilities of neural networks. However, most existing learning-based image compression approaches employ customized convolutional neural network (CNN) to utilize visual features by treating all pixels equally, neglecting the effect of local key features. Meanwhile, the convolutional filters in CNN usually express the local spatial relationship within the receptive field and seldom consider the long-range dependencies from distant locations. This results in the long-range dependencies of latent representations not being fully compressed. To address these issues, an end-to-end image compression method is proposed by integrating graph attention and asymmetric convolutional neural network (ACNN). Specifically, ACNN is used to strengthen the effect of local key features and reduce the cost of model training. Graph attention is introduced into image compression to address the bottleneck problem of CNN in modeling long-range dependencies. Meanwhile, regarding the limitation that existing attention mechanisms for image compression hardly share information, we propose a self-attention approach which allows information flow to achieve reasonable bit allocation. The proposed self-attention approach is in compliance with the perceptual characteristics of human visual system, as information can interact with each other via attention modules. Moreover, the proposed self-attention approach takes into account channel-level relationship and positional information to promote the compression effect of rich-texture regions. Experimental results demonstrate that the proposed method achieves state-of-the-art rate-distortion performances after being optimized by MS-SSIM compared to recent deep compression models on the benchmark datasets of Kodak and Tecnick. The project page with the source code can be found in <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://mic.tongji.edu.cn</uri> .